D(v) is the damping matrix which includes only terms of quadratic drag:

D(v) = -diag{Xu|u| lul, Yv|v| M, Zw|w| lwl ,Kp|p| |p|, Mq|q| M ,Nr|r| M

where, -Xuu is the coefficient of the drag which the vehicle experience due to the motion along the x-axis, and -Nn is the coefficient of the hydrodynamic torque due to the rotational motion of the vehicle with respect to the z-axis. Other coefficients can be described similarly.

The effect of environmental disturbances such as waves and ocean currents are neglected in this study. As a result, following six differential equations describing the 6-DOF equations of motion in surge, sway, heave, roll, pitch, and yaw, respectively, are obtained:

m(u + qw-rv) - X uuu|u| = Fx m(v + ru-pw) - Y vvv|v| = 0 m(w + pv-qu) - Z www|w| = 0 Zxp +- (Zz-Iy ) qr-Kppp|p| = k1Fx !yq +(Ix-Iz ) rp-Mqqq|q| = °

The main thrust is denoted by Fx, thus, the propulsion forces and moments vector can be written as i=[Fx,0,0,kiFx,0,0]T, where ki is a coefficient relating the ratio of the thrust to the rolling moment of the body. The load torque of the propeller causes the body to react with an equal torque and to rotate in opposite direction. The reaction torque is another control input to the system, but, obviously, it is dependent on the thrust. Since the mechanical model does not have control fins, (6) does not include any input for control surface forces and moments. In (6), except for the equations in surge and roll, four equations can be identified as second-order nonholonomic constraints which expose the non-integrable velocity relationships. These constraints imply that possible displacements of the body in each direction are not independent, but are mutually connected.

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